3 Complex Numbers Questions - HW Help (1 Viewer)

[iPhone4]

Hi everyone, I received a massively long assignment on C.N. this week and can't seem to get the three questions here (the ones with the black stars next to them). Could anyone help a fellow MX2'er out here? You'll practice your own skills! The image is attached because I CBF to be bothered to type it up on latex, and find it easier to type on word.



THANKYOU!

Edit: Question 2C says: If z=costheta + isintheta, show that for sin(theta/2) cannot equal zero, (what is shown there)

 

[Makematics]

For the first one, let z=x+iy, where x and y are real
(x+iy)^2 +2(x-iy) + 6 =0
x^2 - y^2 +2xyi + 2x -2iy + 6 = 0
Equating imaginary parts, 2xy -2y=0
xy-y=0
y(x-1)=0
x=1, y=0
Equating real parts, x^2 - y^2 + 2x +6=0
Substituting in y =0, x^2 +2x +6=0
(x+1)^2 +4=0
No real solutions.
Substituting in x=1, 1-y^2 +2 +6=0
y^2=9
y=3, y=-3
∴z=1±3i

I think this is right, but not entirely sure if my process was right with the y=0 :/ Well thats my attempt at it

 

[Makematics]

For the second one,
LHS= (1-cosnӨ) - isinnӨ
= 1- [1-2sin²(nӨ/2)] - 2isin(nӨ/2)cos(nӨ/2)
= 2sin²(nӨ/2) - 2isin(nӨ/2)cos(nӨ/2)
= -2i²sin²(nӨ/2) - 2isin(nӨ/2)cos(nӨ/2)
= - 2isin(nӨ/2) [cos(nӨ/2)+ isin(nӨ/2)]
=RHS

 

[iPhone4]

Thankyou so much! Just with Q1, why isn't it z=1+3i, 1-3i, 1+0i? Because you retrieved y=0 as a soln in the equating of the imaginary bits?

 

[Makematics]

Thankyou so much! Just with Q1, why isn't it z=1+3i, 1-3i, 1+0i? Because you retrieved y=0 as a soln in the equating of the imaginary bits?

No, because if you substitute z=1 back into the original equation you find that it is not a solution. and also, when i found y=0, that had nothing to do with the x=1 part. I'm not even sure if i did it right, but the fundamental theorem of algebra states that there are two solutions to the equation, and i got two which both work when substituted into the equation, so it should be fine

 

[Makematics]

for question 2(c)

Write out what u found in part (b)

Sub in z = cosθ + isinθ

Group the reals, group the imaginaries (factorise i out)

Now u would know what to do

And to show that sin(θ/2) =/= 0, observe the the RHS of what u just shown (i.e. BLAH/sin(θ/2))

I think he meant the question tells you sin(θ/2) =/= 0... but not sure. btw how will he find the answer in part (b) :/ im not sure how to do that one. i dunno if it requires sums of APs or something cos i havent learn that.

 

[HeroicPandas]

I think he meant the question tells you sin(θ/2) =/= 0... but not sure. btw how will he find the answer in part (b) :/ im not sure how to do that one. i dunno if it requires sums of APs or something cos i havent learn that.

yeh u are correcto! i misread it haha

its a geometric progression!



OR

 

[Makematics]

yeh u are correcto! i misread it haha

its a geometric progression!

hmm alright well im gonna leave it to someone else because im seriously unfamiliar with that stuff. now im defs gonna learn it because i can see it can come handy in 4 unit

 

[Makematics]

yeh u are correcto! i misread it haha

its a geometric progression!



OR

Ah yes I forgot about that! damnit! good job. OP do you still need help with the last question?

 

[HeroicPandas]

Hi everyone, I received a massively long assignment on C.N.

Ah yes I forgot about that! damnit! good job. OP do you still need help with the last question?

lol

the last question (2(c)) requires a useful tool in complex numbers and he must see it himself or he wont do well

 

[Makematics]

lol

the last question (2(c)) requires a useful tool in complex numbers and he must see it himself or he wont do well

Yep true, but you gave him a hint before. did you delete it? :S

 

[iPhone4]

Yeah I have no idea about how to do 2c. HALP! haha. Thanks.

 

[HeroicPandas]

Yep true, but you gave him a hint before. did you delete it? :S

yep